Question

Acest exercițiu.Vă mulțumesc !​

Answer 1

Fiind vorba despre logaritmi avem urmatoarele conditii:

$\boxed{x,\ y > 0} \\\\x-2y > 0\iff \boxed{x > 2y}$

$\displaystyle 2\log_2(x-2y)=\log_2x+\log_2y \iff \log_2(x-2y)^2=\log_2xy \\\\\iff (x-2y)^2=xy\iff x^2-4xy+4y^2=xy\\\\\iff x^2-5xy+4y^2=0 \big|:y^2\\\\\implies \cfrac{x^2}{y^2}-5\cfrac{xy}{y^2}+4\cfrac{y^2}{y^2}=0 \iff \left(\cfrac{x}{y}\right)^2-5\cfrac{x}{y}+4=0\\\\\cfrac{x}{y}=t\implies t^2-5t+4=0\\\\\implies (t-1)(t-4)=0\\\\\implies t_1=1 \implies \cfrac{x}{y}=1\implies x=y\implies x < 2y\ contradictie\\\\\ \ t_2=4\implies\boxed{\cfrac{x}{y}=4}$